1. Field of the Invention
This invention relates to a method of designing a lens system or an optical system. Further, this invention relates to a design of diffractive optical elements.
2. Introduction
Design of a lens requires setting up equations defining the relations among optical parameters of the lens, solving the equations and obtaining solutions for determining the parameters. In many cases, the set of equations cannot be solved exactly, because some equations are non-linear or too complicated. The set of equations often leads to a plurality of solutions which contain errors. When the equations are solved, the solution must be estimated by some method, whether or not the solution is valid. A xe2x80x9cmerit function(or cost function)xe2x80x9d is sometimes adopted for estimating the validity of the solution. The merit function is defined as a sum of squares of some errors, for example, a sum of position errors or wavefront errors at points in an imaging region. These errors, termed xe2x80x9caberration errorsxe2x80x9d, appear only in calculation. If aberration errors at individual points are smaller, the merit function is also smaller. Then, a smaller merit function means smaller errors in the solution as a whole. If the merit function is the smallest, the aberration errors should be the smallest. The parameters of lens assemblies or optics are designed by the merit function yielding the minimum value. The function can estimate the validity of the solution as designed parameters. The solutions yielding the merit function of minimum value should realize the most suitable parameters.
In addition to the aberration errors of the solution on the optical equations, production errors appear when the lens assemblies or optical parts are actually manufactured. Production errors hinder the manufacturer from making a lens or optical part having the exact parameters just given by the solution. A production error is defined to be a difference between the designed (calculated) value and the actual value of the product. For simplicity, the word xe2x80x9clensxe2x80x9d is used to express both a xe2x80x9clensxe2x80x9d and an xe2x80x9coptical partxe2x80x9d hereafter. A large production error degrades the produced lens and sometimes segregates the produced lens into a classification of inferior goods. Allowable scope of manufacturing errors is beforehand determined for satisfying the requisites for the lens. The maximum of an allowable production error is called xe2x80x9ctolerancexe2x80x9d. A large tolerance facilitates production; it is easier to manufacture a lens which is defined by parameters with bigger tolerances. A small tolerance imposes a heavy burden on the manufacturer; it is difficult to make a lens having designed parameters with small tolerances. Thus, tolerance is a measure of ease of production.
A solution gives a set of optimum values and tolerances of the parameters. Although a solution gives excellent performance to the lens having the exact parameters which are equal to the solution values, the solution is not necessarily the best solution. If the parameter tolerances of the solution are small, it is difficult to make lens having errors of parameters within tolerance. The performance of the lens which has parameters equal to the designed values is called the xe2x80x9cbest performancexe2x80x9d for the solution. Even if a solution has an excellent and best performance, the solution is not an optimum solution if tolerances are narrow. People believe that the best solution is a solution which gives the highest performance to the product, but this is not necessarily true. If tolerances are small, production is difficult, even though the solution gives the best performance. The best solution is not the solution giving the best performance but should be the solution which gives xe2x80x9cwide tolerancesxe2x80x9d as well as xe2x80x9cbest performancexe2x80x9d. Wide tolerance is more important than best performance. A purpose of the present invention is to provide a method of designing lens assemblies or optical parts which gives parameters large tolerances for facilitating production.
Words are clarified by defining the exact meanings. There are various parameters which define lenses or optical parts. The parameters can be classified by two standpoints. One standpoint is classification into the parameters which are treated as variables in calculation seeking optimum designs for lenses and into the parameters which are treated as constant values in the same calculation. In the case of designing an optical system having a plurality of lenses, variable parameters are, e.g., the thicknesses of lenses, the curvatures of both surfaces of the lenses, and the distances between the lenses for which calculation is done for seeking optimum values which satisfy the required conditions. Other parameters are treated as constants keeping predetermined values in the calculation. For example, constant parameters are the distance between the light source and the lens, the thickness and the curvature (=0) of a window, the shape of some lenses and the distance between selected lenses. The physical constants, for instance, refractive index of lenses or dispersion are treated as constant parameters in the calculation, since they are previously determined by the materials of the lenses. The number of lenses is also a constant parameter, when the number is preliminary determined. The predetermined requirements assign some parameters either to variable parameters or constant parameters. Thus the number of lenses or the material of lenses can be a variable parameter in other case which allows the material and the number to change. Thus, the distinction between variable parameters and constant parameters is the first standpoint of classification.
The other classification of parameters is the parameters to which allocated-errors are given and the parameters to which allocated-errors are not given. The xe2x80x9callocated-errorxe2x80x9d is not a known concept but is a quite novel concept. The allocated-errors play a central role in the present invention. The classification of parameters by the allocated-error is a key idea of the invention. Above explanation of parameters clarifies the first classification into variable parameters and constant parameters and the second classification into error-allocated parameters and non-error-allocated error parameters. Another important distinction relates to the kinds of errors. There are three errors for a parameter: the first one is an aberration-error, the second is a production error, and the third an allocated-error.
All the parameters have production errors which are the deviations of parameters of the product from the parameters given by the solution. Production errors accompany both variable parameters and constant parameters. A solution gives optimum values for variable parameters, for instance, thicknesses of lenses, curvatures of surfaces and distances between lenses. When a manufacturer produces an optical part, the variable parameters deviate from the designed values. The deviations are the production errors of variable parameters.
Constant parameters which are preliminary determined are also suffering from production errors. Thus, there are extra parameters which exclusively denote production errors themselves. Wedge, decenter, tilt, surface irregularity, and refractive index non-uniformity(inhomogeneity) are the words signifying production errors which should be 0 in an ideal product. Design of lenses premises that the errors are 0. Then, these parameters can be named error parameters. Error parameters are defined as differences between constant parameters and the actual values of a product. Error parameters accompany not variable parameters but constant parameters. Wedge denotes an inclination between a front surface and a rear surface of a lens. Decenter means a vertical difference between central axes of lenses. Tilt is an inclination of a lens to a plane perpendicular to the axis. Surface irregularity is a deviation of a product surface from a designed surface. Non-uniformity of refractive index denotes the spatial fluctuation of refractive index of a lens. This invention intends optimization processing by selecting parameters suffering from large production errors among all parameters and positively giving errors to the selected parameters. One feature of the invention is positively to allot errors to parameters. The parameters to which errors are allotted are called error-allocated parameters. The error-allocated parameters can be either the variable parameters which are treated as variables in calculation or the constant parameters which are treated as constants. Further the error parameters, e.g., wedge, decenter, tilt and so on can be the parameters to which errors are assigned. Namely, all three kinds of parameters, i.e., variable parameters, constant parameters and error parameters, can be candidates of error-allocated parameters.
If a parameter P is allocated with errors xc2x1xcex4, the parameter comes to have three values, Pxe2x88x92xcex4, P and P+xcex4. Namely, the parameter has the maximum value P+xcex4 and the minimum value Pxe2x88x92xcex4 as well as the middle value P. Such an allotment of errors is common both to the constant parameters and the variable parameters. In the case of a variable parameter, when the optimum processing calculations change the value of the parameter, the errors xc2x1xcex4 will be allocated to a new value Pxe2x80x2. Then the parameters are Pxe2x80x2xe2x88x92xcex4, Pxe2x80x2 and Pxe2x80x2+xcex4. When thickness of a lens is a variable parameter which should be allocated with errors xc2x10.5 mm and the thickness happens to be 10 mm in calculation, the thickness should be treated as a parameter having three values 10 mm, 9.5 mm and 10.5 mm. When the thickness is changed from 10 mm to 11 mm in the series of calculations, the thickness will have the three values of 10.5 mm, 11 mm and 11.5 mm by allotting xc2x10.5 mm errors. Parameters can be classified into four categories with regard to allocation of errors, as follows:
1. error-allocated variable parameter
2. error-allocated constant parameter
3. non-error-allocated variable parameter
4. non-error-allocated constant parameter
Optical elements mean optical devices which refract, allow to pass, absorb, converge, reflect, diffuse or diffract light beams. The word xe2x80x9cdiffractivexe2x80x9d is a contrary concept of xe2x80x9crefractivexe2x80x9d. Refraction denotes bending of light beams by a difference of refractive indices between air and transparent media (lenses or prisms). In refraction, Snell""s law determines the bending angles of beams at interfaces between air and lenses or prisms. Since all the beams are considered to be refracted by the transparent media individually, the refraction can fully be treated by geometric optics. Geometric optics treats individual beams as making their own different ways in media and progressing along straight lines in homogeneous media. In refraction, it is possible to trace individual light beams. Tracing of individual beams enables the geometric optics to calculate modes of convergence or divergence of the beams. Geometric optics does not treat light beams as waves but treats light beams as an assembly of rays for considering refraction. Geometric optics further does not take phases of waves into account. Sometimes reflection is opposed to refraction. However, the refractive index also rules reflection. Thus, reflection is not a contrary concept to refraction. The geometric optics can handle modes of convergence in a reflection telescope.
On the contrary, a diffractive optical element (DOE) is a new optical device for accomplishing some functions by utilizing diffraction phenomena of light. Light is not an assembly of rays but an assembly of waves. Light should be considered as a packet of waves with phases. Geometric optics is in vain for diffraction. Instead, wave optics can treat diffraction phenomena. The concept of rays is of no use. Diffraction forces us to consider light not as progressing beams but as oscillating waves having continual wavefronts. A diffraction grating is a well-known device making use of diffraction.
There are both a reflection type and a penetrating(transparent) type of diffraction grating. A diffraction grating has many parallel grooves formed with a common period on a substrate. The diffraction grating disperses white light into various colors in different directions like a prism. Diffraction gratings, useful in spectroscopy, can divide monochromatic light into the 0-th order diffraction, the plus and minus first order diffraction, the plus and minus second order diffraction and so on. Bragg""s condition d sin xcex8=mxcex determines the mode of diffraction, where d is the period of grooves, xcex is the wavelength of the monochromatic light, m is the order of diffraction and xcex8 is the direction of the diffracted light. Diffracted light accompanies all the directions xcex8 defined by Bragg""s condition. But an increase of the diffraction order m rapidly decreases the intensity of diffracted light. The conventional diffraction grating has only a function of separating a monochromatic plane wave into an indefinite number of parallel rays which differ in angle vertical to the direction of grooves. When the direction of propagation is the z-direction and the direction of grooves is the y-direction, diffraction makes y-parallel lines which disperse in the x-direction on an image plane. The diffraction grating is two-dimensional and symmetric in the function, since periodical, parallel grooves induce diffraction according to the symmetric Bragg""s condition. The conventional diffraction gratings have only restricted effects since the equivalent grooves have been dug at equal intervals on a substrate uniformly. The conventional diffraction gratings are impotent for controlling of high freedom.
Diffractive optical elements (DOEs) are new devices having a wide and rich scope of functions in advance of the classical diffraction gratings. DOEs are sometimes called holographic optical elements. Instead of parallel grooves, a DOE has two-dimensionally distributed protrusions or cavities which are made on many unit cells defined on the element surface. One purpose for DOEs is the production of equi-parted, equivalent K convergence spots (1xc3x97K) on the image plane. Other purposes are (1) the production of equi-parted, equivalent Kxc3x97L converging spots in K lines by L columns on the image plane, (2) the production of arbitrarily-parted, equivalent Kxc3x97L converging spots on the image plane, and (3) the yield of some character as an assembly of converging spots on the image plane. In any case, the diffractive optical element (DOE) can directly divide the light power from a single light source into many light beams at most.
The formation of a single clear image having a single converging point is the purpose of refractive type and reflective type optical elements, that is, lenses or mirrors. The diffractive optical elements make a plurality of convergences and pay little attention to the formation of images. Then, DOEs would have many utilities different from the refractive or reflective optical elements.
A CO2 laser emits strong infrared coherent light of 10.6 xcexcm. CO2 lasers have widely been used for welding, cutting and annealing of metals. A novel, promising use of the CO2 laser will be the drilling of many narrow holes (about 100 xcexcm diameter) on a printed circuit board. Circuit board hole perforation is at present done by mechanical drilling in which a narrow needle is rotated. Another apparatus of perforating holes on a circuit board is a combination of a CO2 laser, a galvanometer oscillating a small mirror for deflecting reflected light beams and a special f-xcex8 lens converging the oscillating beam on spots of the circuit board. An ordinary lens converges xcex8-slanting beams at a spot distanced by f tan xcex8 from the center on the image plane. On the contrary, the f-xcex8 lens converges xcex8-slanting beams at a point distanced by xcex8 from the center on the image plane. Swaying the galvano-mirror at angles of 0, xc2x1xcex8, xc2x12xcex8, xc2x13xcex8 . . . perforates many small holes distributed in a line with a definite interval on the circuit board. The utilization of two galvanomirrors enable the laser drilling machine to perforate holes at arbitrary spots by scanning the beams two-dimensionally.
The holes are individually bored one by one by the single laser beam. However, the laser-galvanometer-fxcex8-lens apparatus can bore holes on circuit boards at higher efficiency, than mechanical drilling, since the galvanometer oscillates the mirror at a higher speed than the drilling tool. Optical boring would be less expensive than mechanical boring. However, the inertia on the galvano-mirror restricts the upper limit of the oscillation of the galvanometer. Besides, it is still difficult to produce a f-xcex8 lens with high accuracy.
Then, the Inventor hit upon an idea of the use of diffractive optical elements for boring holes on a printed circuit board at one time instead of mechanical drilling and laser-galvanometer boring. Diffractive optical elements would be able to bore all the necessary holes of Kxc3x971 at a stroke which are bored by a single sway of the galvano-mirror. Furthermore, the DOE may be able to perforate a two-dimensional array of holes at a once without movement of parts. If a DOE could divide a laser light into partial Kxc3x97L beams, the divided partial beams would bore Kxc3x97L holes on a print circuit board at one time. The DOE simultaneous perforation will dispense with the galvanometer and the f-xcex8 lens.
This application claims the priority of Japanese Patent Applications No. 10-69480(69480/1998) filed Mar. 3, 1998 and No.11-34012(34012/1999) filed Feb. 12, 1999 which are incorporated herein by reference.
FIG. 1 shows a scheme of designing an optical part containing lenses. The first step of designing of an optical part is the determination of initial values (starting values) of the lens material (refractive index, dispersion, absorption coefficient), the number of lenses, surface shape (sphere or asphere), relative disposition (lens-lens distance, lens-image distance) and the determination of variable parameters. Namely, parameters are divided into constant parameters and variable parameters at the first step. This is the fundamental structure setting. Besides the parameters assigned with initial values, there are some additional conditions, e.g., the wavelength of the light source, the thickness of the lens, the material, the full size of the part and so on. Optical equations including the parameters should be set up from the relation of the refraction of beams on surfaces of lenses by taking account of the requirements. Then, the equations should be solved, and the solution is a set of candidate parameters, e.g., lens thicknesses, curvatures, aspherical coefficients, distances. A set of candidate parameters can define an optical part. The solution contains a set of candidate parameters which enables a designer to define a candidate optical part.
In some cases, the optical equations are too abstract to determine a unique solution. Although the requirements settle extra equations, the number of equations is still less than the number of the variable parameters in many cases. A single, unique solution cannot be obtained due to the unsufficient number of equations. The equations are often non-linear. In the case of adopting aspherical surface lens assemblies, the surface itself has too many parameters for defining the surface profile. Owing to too many parameters, the set of equations cannot be analytically solved. In these cases, equations are forcibly solved by employing various approximation methods or ray tracing method which yield a plurality of candidate solutions. There are often an indefinite number of approximation solutions which satisfy the equations defining the refraction or reflection on the surfaces of lens assemblies or mirrors. It is a rare case that an initially-attained solution yields a set of optimum parameters. Since the set of optical equations allows many solutions, the designer should determine the most suitable solution from the many candidate solutions which satisfy the equations approximately.
The validity of solutions is examined by a xe2x80x9cmerit functionxe2x80x9d. A merit function is defined as a sum of squares of errors. Lower errors decrease the merit function. The lower the errors are, the higher the performance is. Thus, the merit function is a measure of estimating the performance of the product which has the parameters the solution gives. Furthermore, minimizing the merit function can determine the suitable parameters. The merit function is a sum of squares of optical errors (aberration) multiplied by some weights at several points. Of course, there are many kinds of optical errors (aberration). The merit function is defined by adopting some pertinent kinds of errors. For example, the merit function for a lens assembly employs the aberration of wavefronts or the ray aberration which is the deviation of rays from a focus on an imaging plane. The merit function chooses a suitable aberration as the errors at estimation points in accordance with this purpose.
FIG. 2 shows an ideal set of wavefronts made by a lens which converges a plane wave at a focal point. The wavefronts are parallel planes in an ideal plane wave. When an ideal lens converges the plane wave, the wavefronts come to a spherical wave having concentric sphere wavefronts. However, when an actual wave is converged by a lens, the wavefronts often deviate from the ideal spheres. FIG. 3 denotes the deviation of wavefronts. An arc is an ideal wavefront made by an ideal lens from an ideal plane wave. A wavy curve denotes an actual wavefront produced by an actual lens having some aberration. The wavefronts are different for the ideal lens (arc) and the actual lens (wavy curve). The wavefront aberration is the difference between an ideal wavefront and an actual wavefront. The wavefront aberration deforms an image made by a lens at the focal point. An image which should be a circle is deformed into non-circle. The spot diameter at the focus is wider than that of the ideal image. The deviation of an image from a circle is one kind of error. The deviation of the spot diameter is another kind of error. Other kinds of errors can be defined for deviations of physical parameters. Any selection of the kinds of errors can construct a merit function.
Calculation of the equations makes a plurality of candidate solutions. A solution determines a set of the values of parameters that enable the designer to calculate the actual values of the errors. The values of the errors yield the merit function an actual value that is a measure of the validity of the solution. Merit functions are calculated for a plurality of solutions. The optimum design is given by the solution which has the minimum merit function among the solutions called an optimizing calculation. The parameters given by the optimum solution are named optimum parameters. For example, aberration coefficients sj are assumed for signifying some aberration s at point j. And sj0 is an ideal coefficient of sj, with weight denoted by wj. A merit function, expressed as "PHgr"=xcexa3wj(sjxe2x88x92sj0)2, is defined as a sum of squares of the errors and is minimized for seeking the optimum solution. Calculating errors from a solution, summing squares of the errors, minimizing the sum and estimating the solution are the inherent, conventional role of a merit function.
However, the merit function can be used in another manner. Unlike the conventional method, the processing starts from the merit function. Instead of starting the equations, calculation begins on the merit function. An approximation method assumes a form of a merit function, gives initial values of optical parameters (refractive index, thickness, curvature, aspherical coefficient), calculates the merit function, changes the values of parameters in the direction of reducing the merit function and attains the least merit function. Without solving the equations, an assumption of initial values of the parameters enables the designer to take the ray tracing method and calculate the ray aberration or the wavefront aberration. The assumed initial values give the wavefront aberration and the ray aberration, from which the merit function can be calculated. Instead of solving the equations, a candidate answer is assumed from the beginning. The ray tracing method is carried out on the assumed parameters. By tracing all the relevant rays, the wavefronts and the spots on an imaging plane are calculated. Then, the wavefront aberration and the ray aberration can be obtained and the merit function is calculated. The result is an estimation of the assumed solution. Many candidate answers (solutions) can be produced by adding small increments to the parameters of the initial solution. A plurality of the candidate solutions are assumed one by one and estimated by calculating the merit functions of the candidates and by comparing the values of the merit functions.
Comparison of the values of the merit function clarifies the set of parameters minimizing the merit function. This is the optimizing calculation. The optimum solution enables the designer to trace the rays, to calculate the wavefronts and to simulate the optical properties of the designed lens. Besides, the degeneration of performance is investigated by giving the parameters tentative errors from the optimized values positively. An increase of tentative errors reduces the performance in general. The fall in performance is small for small errors. Large errors reduce the performance to a great extent. The scope of an allowable fall is determined by considering the performance.
An error which degrades the performance within the allowable level is ignored. But an error which abases the performance beyond the allowable level should be forbidden. The error which degenerates the performance just to the allowable level is the maximum allowable error, called a tolerance. One parameter has one tolerance. All tolerances are calculated for all parameters as the errors which reduce the performance just to the allowable level. This is the tolerancing(tolerance analysis). The fundamental structure setting, the optimizing calculation, simulation and tolerancing build the optical design in FIG. 1.
Sample lens assembly or optical parts are actually produced tentatively from the result of the optical design. The sample has all the parameters based upon the solution obtained by the optical design. The production of samples is the trial production (prototyping). The actual samples are estimated from the standpoint of actual performance, production cost and production feasibility. This is the trial production estimation. The former merit function has estimated the optical parts by taking only the performance into account, ignoring the cost and feasibility of production. Thus, the trial production estimation is indispensable. The performance, the cost and the feasibility are the three criteria in the trial production estimation.
When the result of the trial production estimation is satisfactory, the lens design finishes successfully. The parameters determined by the design should be employed in actual mass production. If the result of the trial production estimation is unsatisfactory, the design should return to the first step of the fundamental structure setting of the optical design in FIG. 1. Then the same steps should be repeated from the fundamental structure setting. Namely, another candidate solution is tentatively made by assuming another set of initial parameters. The optical design process repeats the optimizing calculation and the simulation. Then the trial production estimation follows the simulation. If the result is unsatisfactory, the same processes should be further repeated. Similar processes including the optical design and the trial production estimation should be repeated until the trial products satisfy the required performance.
It takes a long time to carry out even a series of the optical design and the trial production estimation. Furthermore, repetitions of similar processing consume much time which is a product of the one-cycle time multiplied by the number of repetitions. Then, when the number of repetitions is large, development expense rises enormously and development time increases, which are not desirable. A new way to find optimum solutions faster is desired.
Instead of geometric optics, wave optics gives a clue of designing diffractive optical elements (DOEs). The conventional design of DOEs is now outlined. A diffractive optical element (DOE) produces a desired pattern on an image plane from monochromatic laser light by phase modulating the monochromatic light by a fine stepped pattern on the surface of the element. In the case of refractive or reflective optical elements, that is, lenses or mirrors, beams are traced by geometric optics which deems light as an assembly of rays. Instead, DOEs are designed by Fourier optics analysis based upon diffraction theory. A fundamental optical system including a DOE is shown by FIG. 22. A plane wave goes into and passes a diffractive optical element (DOE). The DOE phase modulates the plane wave. The phase-modulated wave is converged by a convergence lens and imports an image pattern on an image plane. A three-dimensional coordinate is defined on the diffraction system, where the propagating direction is determined to be the z-direction. The planes vertical to the z-direction are xy-planes. The surfaces of the DOE and the image plane are in xy-planes. A laser (not shown in FIG. 22) emits a monochromatic plane wave exp(jkz), where k is a wavenumber(k=2xcfx80/xcex). In fact, the actual light emitted by a laser is not a rigorous plane wave but some distribution in xy-plane. Thus, the amplitude of the incident light at a point (x,y) on the element is now denoted by a(x,y). Thus, the laser light is expressed by a(x,y)exp(jkz). Since the DOE is an element for inducing diffraction, monochromaticity is important. Laser light is pertinent as the light source due to the excellent monochromaticity. However, it is also possible to analyse the non-monochromatic light, because the diffraction on the image plane can simply be produced by superimposing the images for different wavelengths.
Here, the light source is a CO2 laser which makes monochromatic light of 10.6 xcexcm. The complex amplitude transmittance of the DOE is denoted by t(x,y). When a unit intensity 1 of light enters a front surface point (x,y) of the DOE, light of amplitude transmittance t(x,y) goes out from a rear surface point (x,y). The xe2x80x9camplitude transmittancexe2x80x9d means that t(x,y) includes phase in addition to intensity. The complex amplitude transparency t(x,y) can be expressed simply by expand). Here, xe2x80x9cdxe2x80x9d is the thickness of the DOE at point (x,y) and xe2x80x9cnxe2x80x9d is the complex refractive index of the DOE. The complex amplitude of the light at the back of the DOE is given by a(x,y)t(x,y). A converging lens follows the DOE for making a clear image at an image plane. The DOE and the lens modulate the laser light a(x,y) into a complex amplitude U(x,y) at the rear surface of the lens. U(x,y) shall be a product of a(x,y), t(x,y) and a lens factor.
U(x,y)=a(x,y)t(x,y)exp{xe2x88x92jk(x2+y2)/2f}xe2x80x83xe2x80x83(1)
Here, it is assumed that the lens has a sufficiently wide aperture and is free from aberration. Apparently, Eq. (1) does not include the refractive index of the lens. The refractive index of the lens is not insignificant. The refractive index of the lens is implied in the focal length f. P(x,y) is an arbitrary point on a lens middle plane, O is the center of the lens and F is the focal point on the image plane. The partial waves passing any points in the lens have all the same phase at the focus F, because F is the focal point. The phase difference between partial waves passing O and passing P originates from the path difference between PF and OF. The path difference (PF-OF) is (x2+y2+f2)xc2xdxe2x88x92f, since PF=(x2+y2+f2)xc2xd and OF=f. Approximately, the difference is (x2+y2)/2f Phase difference is a product of the path length and the wave number k. Since this consideration is retrospective in time, a minus sign xe2x80x9cxe2x88x92xe2x80x9d should be attached. The last term of Eq. (1) means the phase change due to the phase delay of the partial wave which passes the lens at point P. U(x,y) is the complex amplitude just at the rear surface of the lens. The image plane is distanced by f from the lens, where xe2x80x9cfxe2x80x9d is the focal length of the lens. W(X,Y) is the complex amplitude on the image plane. Two different coordinates shall be discriminated. The coordinates (x,y) in small letters mean the positions on the lens. The coordinates (X,Y) in capital letters mean the positions on the image plane. The distance between a lens point (x,y) and an image point (X,Y) is {(Xxe2x88x92x)2+(Yxe2x88x92y)2+f2}xc2xd. The path difference xcex94 between the arbitrary partial wave and the axial partial wave OF is xcex94={(Xxe2x88x92x)2+(Yxe2x88x92y)2+f2}xc2xdxe2x88x92f. Approximately, xcex94={(Xxe2x88x92x)2+(Yxe2x88x92y)2+f2}/2f. Multiplying xcex94 by k gives the phase difference kxcex94=k{(Xxe2x88x92x)2+(Yxe2x88x92y)2}/2f. The change of amplitude is exp(jkxcex94). The complex amplitude W(X, Y) on the image plane is a sum of infinitesimal contributions U(x, y)exp(jkxcex94)dxdy from lens point (x, y). W(X, Y) can be obtained by integrating U(x, y)exp(jkxcex94)dxdy on the lens surface.                               W          ⁡                      (                          X              ,              Y                        )                          =                  exp          ⁢                      {                                                            j                  ⁢                  k                                ⁡                                  (                                                            X                      2                                        +                                          Y                      2                                                        )                                                            2                ⁢                f                                      }                    ⁢                      ∫                          ∫                                                U                  ⁡                                      (                                          x                      ,                      y                                        )                                                  ⁢                exp                ⁢                                  {                                                                                    j                        ⁢                        k                                            ⁡                                              (                                                                              x                            2                                                    +                                                      y                            2                                                                          )                                                                                    2                      ⁢                      f                                                        }                                ⁢                exp                ⁢                                  {                                                            -                                                                        j                          ⁢                          k                                                ⁡                                                  (                                                      xX                            +                            yY                                                    )                                                                                      f                                    }                                ⁢                                  xe2x80x83                                ⁢                                  ⅆ                  x                                ⁢                                                      ⅆ                    y                                    .                                                                                        (        2        )            
Substitution of Eq. (1) for U(x,y) of Eq. (2) brings about W(X,Y),                               W          ⁡                      (                          X              ,              Y                        )                          =                  exp          ⁢                      {                                                            j                  ⁢                  k                                ⁡                                  (                                                            X                      2                                        +                                          Y                      2                                                        )                                                            2                ⁢                f                                      }                    ⁢                      ∫                          ∫                                                a                  ⁡                                      (                                          x                      ,                      y                                        )                                                  ⁢                                  t                  ⁡                                      (                                          x                      ,                      y                                        )                                                  ⁢                exp                ⁢                                  {                                                            -                                                                        j                          ⁢                          k                                                ⁡                                                  (                                                      xX                            +                            yY                                                    )                                                                                      f                                    }                                ⁢                                  ⅆ                  x                                ⁢                                                      ⅆ                    y                                    .                                                                                        (        3        )            
Eq. (3) includes double integration by x and y on the lens surface. Two functions of u and v are introduced for simplifying the expression of W(X,Y). A(u,v) is a Fourier transformation of a(x,y). T(u,v) is a Fourier transformation of t(x,y). Namely,
T(u,v)=∫∫t(x,y)exp{xe2x88x92j(ux+vy)}dxdy.xe2x80x83xe2x80x83(4)
A(u,v)=∫∫a(x,y)exp{xe2x88x92(ux+vy)}dxdy. By T(u,v) and A(u,v), the complex amplitude W(X,Y) on the image plane is simplified into                               W          ⁡                      (                          X              ,              Y                        )                          =                  exp          ⁢                      {                                                            j                  ⁢                  k                                ⁡                                  (                                                            X                      2                                        +                                          Y                      2                                                        )                                                            2                ⁢                f                                      }                    ⁢                      A            ⁡                          (                                                kX                  f                                ,                                  kY                  f                                            )                                ⁢                                    T              ⁡                              (                                                      kX                    f                                    ,                                      kY                    f                                                  )                                      .                                              (        5        )            
The pattern of the DOE determines T(u,v). The intensity profile of the laser determines A(u,v) in Eq. (5). Indeed, the final complex amplitude on the image plane depends also upon the light source profile a(x,y). In an actual calculation, W(X,Y) should take a(x,y) into account. However, here a(x,y) is assumed to be constant (a(x,y)=1), which assumes the incident light is a perfect plane wave, for clarifying the relation between the diffraction image W(X,Y) and the DOE pattern T(u,v). Since the actual intensity is a square |W(X,Y)|2 of amplitude W(X,Y), the coefficient exp{jk(X2+Y2)/2f} will vanish. For a perfect plane wave, the complex amplitude W(X,Y) on the image plane is,
W(X,Y)=T(kX/f, kY/f).xe2x80x83xe2x80x83(6)
Eq. (6) is an approximated expression for simplifying the following explanation. In practice, Eq. (5) should be used for the calculation of W(X,Y). Now, Eq. (6) means that the diffracted pattern W(X,Y) on the image plane is entirely equivalent to the Fourier transformation T(kX/f, kY/f) of the DOE pattern t(x,y). Eq. (6) can be interpreted as that the DOE pattern t(x,y) determines the diffraction image W(X,Y). However, in actual cases, an object W(X,Y) is given first of all. Then, the corresponding pattern t(x,y) of the DOE is sought for the W(X,Y). For example, in the case of producing one-dimensionally equi-parted K converging spots aligning on a direct line on an image plane, W(X,Y) is determined to take a definite value at K spots aligning on the direct line and to take value 0 at all the other points. In the case of making two-dimensionally equi-parted Kxc3x97L spots in a rectangular area on an image plane, W(X,Y) should take a definite value at Kxc3x97L spots in the corresponding area and value 0 at all the other points. The desired pattern on an image plane determines W(X,Y). The problem is what DOE will make W(X,Y) on the image plane. Since W(X,Y) is connected to t(x,y) by Eq. (3) and Eq. (5) or Eq. (6), inverse Fourier transformation of W(X,Y) would simply make the pattern t(x,y) on the DOE.
However, this Inventor does not choose the inverse Fourier transformation from W(X,Y) for the sake of the difficulty of manufacturing. The inverse Fourier transformation determined desirable phase distribution t(x,y) on a DOE, t(x,y) would not be cell-discrete functions but would be a continual function. Continually phase-changing elements are difficult to manufacture. A DOE which is not divided into discrete cells is unsuitable for production due to the irregularity of individual concavities or convexities. A DOE should have such a pattern that is defined upon discrete unit cells. The concaves and concavities on the DOE cannot realize the amplitude distribution. Endowment of the amplitude distribution requires the DOE of transmittance distribution onto the DOE. When the DOE had transmittance distribution, the light energy which is absorbed or reflected by the transmittance fluctuation would be dissipated as a loss. Further, the phase distribution would have continual, non-discrete values which a step-wise DOE could not realize. Hence, the inverse Fourier transformation should be abandoned and instead, calculation shall proceed from a cell-discrete DOE to an image pattern.
FIG. 23 shows an example of a DOE pattern in which a DOE has M cells in the x-direction (horizontal) and N cells in the y-direction (vertical). A unit cell has a width xcex4x in the x-direction and a length xcex4y in the y-direction. The DOE has MN cells. The size of the DOE is M xcex4x in the x-direction and Nxcex4y in the y-direction. The purpose of quantizing a DOE into a lattice structure of discrete cells facilitates both manufacture and calculation. Every cell is allocated with a variable. The variable is the height, since the DOE should induce diffraction by varying the phase with the differences of periodically-changing optical paths. The height of the cells should be binary (two-valued), quadruple (four-valued) or so (2M-valued). In the binary case, the phase should be either 0 or xcfx80. Cell height determines the phase of penetrating or reflecting light. The binary-phase DOE gives binary heights of cells. In the quadruple case, the phase should be 0, xcfx80/2, xcfx80 and 3xcfx80/2. Cells take four kinds of heights. In general, cell heights are 2M in the case of 2M phases. The variable xe2x80x9cheightxe2x80x9d may be called a xe2x80x9cthicknessxe2x80x9d, since a DOE is a board.
The quadruple case is rather complicated. The following steps of design is explained by adopting the binary variable case for simplicity. The phase difference between DOE-penetrating wave and air-passing wave is 2xcfx80(nxe2x88x921)d/xcex, where xe2x80x9cnxe2x80x9d is the refractive index, d is the thickness and xcex is the wavelength of the light. A binary DOE takes two values d1 and d2 of thickness. The difference (d2xe2x88x92d1) should correspond to the phase difference xcfx80. Namely, 2xcfx80(nxe2x88x921)(df2xe2x88x92d1)/xcex=xcfx80. The DOE material determines the refractive index n. Then, the height difference (d2xe2x88x92d1) is uniquely determined to be (d2xe2x88x92d1)=xcex/2(nxe2x88x921). The difference is determined from xcfx80 phase difference but the thickness d1 itself is arbitrary. The DOE plane is divided into horizontal M cells and vertical N cells. Cell coordinate (m,n), for instance, is defined by posing the origin at the central cell. Horizontal cell number m varies from xe2x88x92M/2 to +M/2xe2x88x921. Vertical cell number n changes from xe2x88x92N/2 to +N/2xe2x88x921. Every cell shall be allocated with a phase. xcfx86mn is the phase of the cell (m,n) which is either 0 or xcfx80. The complex amplitude transmittance is denoted by tmn. When the DOE material has no absorption, the magnitude of tmn is a unit number |tmn|=1.
tmn=exp(jxcfx86mn)=+1, xcfx86mn=0
xe2x88x921, xcfx86mn=xcfx80.xe2x80x83xe2x80x83(7)
The problem is to find out the most pertinent distribution of the cell complex amplitude transmittance {tmn}. Since coordinates x and y are continual but cells are discrete, it is difficult to give a simple expression to t(x,y) from the cell phases. The following rectangular unction rect(x) is defined for reconciling continual (x,y) with cell phases xcfx86mn.
rect(x)=1, |x|xe2x89xa6xc2xd
0, |x| greater than {fraction (1/2)}.xe2x80x83xe2x80x83(8)
t(x,y) is only a sum of complex amplitude transmittance tmn of Mxc3x97N cells. When (mxe2x88x920.5) xcex4xxe2x89xa6x less than (m+0.5) xcex4x and (nxe2x88x920.5) xcex4yxe2x89xa6y less than (n+0.5) xcex4y, t(x,y)=tmn. The same matter can be otherwise expressed by the rectangular function.                               t          ⁡                      (                          x              ,              y                        )                          ⁢                              ∑                          m              =                                                -                  M                                /                2                                                                    M                /                2                            -              1                                ⁢                      xe2x80x83                    ⁢                                    ∑                              n                =                                                      -                    N                                    /                  2                                                                              N                  /                  2                                -                1                                      ⁢                          xe2x80x83                        ⁢                                          t                mn                            ⁢                              rect                ⁡                                  (                                                            x                      -                                              m                        ⁢                                                  xe2x80x83                                                ⁢                                                  δ                          x                                                                                                            δ                      x                                                        )                                            ⁢                                                rect                  ⁡                                      (                                                                  y                        -                                                  n                          ⁢                                                      xe2x80x83                                                    ⁢                                                      δ                            x                                                                                                                      δ                        y                                                              )                                                  .                                                                        (        9        )            
The Fourier transformation of t(x,y) is T(kX/f, kY/f)=W(X,Y) (under some assumptions). This assumes:                               W          ⁡                      (                          X              ,              Y                        )                          =                              ∑                          m              =                                                -                  M                                /                2                                                                    M                /                2                            -              1                                ⁢                      xe2x80x83                    ⁢                                    ∑                              n                =                                                      -                    N                                    /                  2                                                                              N                  /                  2                                -                1                                      ⁢                                          t                mn                            ⁢                              ∫                                  ∫                                                            rect                      ⁡                                              (                                                                              x                            -                                                          m                              ⁢                                                              xe2x80x83                                                            ⁢                                                              δ                                x                                                                                                                                          δ                            x                                                                          )                                                              ⁢                                          rect                      ⁡                                              (                                                                              y                            -                                                          n                              ⁢                                                              xe2x80x83                                                            ⁢                                                              δ                                y                                                                                                                                          δ                            y                                                                          )                                                              ⁢                    exp                    ⁢                                          {                                                                        -                                                      jk                            ⁡                                                          (                                                              xX                                +                                yY                                                            )                                                                                                      f                                            }                                        ⁢                                          ⅆ                      x                                        ⁢                                                                  ⅆ                        y                                            .                                                                                                                              (        10        )            
The range of summation of m is from xe2x88x92M/2 to M/2xe2x88x921. The scope of summation of n is from xe2x88x92N/2 to N/2xe2x88x921. The double integration can be done for every cell separately. The Fourier transformation of a unit cell can be expressed by a sinc function sinc(x).                               sin          ⁢                      xe2x80x83                    ⁢          c          ⁢                      xe2x80x83                    ⁢                      (            x            )                          =                                            sin              ⁢                              xe2x80x83                            ⁢                              (                                  π                  ⁢                                      xe2x80x83                                    ⁢                  x                                )                                                    π              ⁢                              xe2x80x83                            ⁢              x                                .                                    (        11        )            
Sinc(x) is an integral (2xcfx80)xe2x88x921 ∫exp(jhx)dh with a range between h=xe2x88x92xcfx80 and h=+xcfx80. Sinc(x) takes the maximum 1 at the limit x=0. Sinc(x), which is an even function, decreases, waving in both directions. The sinc function gives another expression to W(X,Y). The sinc function serves W(X,Y) with the waving parts.                               W          ⁡                      (                          X              ,              Y                        )                          =                              δ            x                    ⁢                      δ            y                    ⁢          sin          ⁢                      xe2x80x83                    ⁢                      c            ⁡                          (                                                                    δ                    x                                    ⁢                  X                                                  λ                  ⁢                                      xe2x80x83                                    ⁢                  f                                            )                                ⁢          sin          ⁢                      xe2x80x83                    ⁢                      c            ⁡                          (                                                                    δ                    y                                    ⁢                  Y                                                  λ                  ⁢                                      xe2x80x83                                    ⁢                  f                                            )                                ⁢                                    ∑                              m                =                                                      -                    M                                    /                  2                                                                              M                  /                  2                                -                1                                      ⁢                          xe2x80x83                        ⁢                                          ∑                                  n                  =                                                            -                      N                                        /                    2                                                                                        N                    /                    2                                    -                  1                                            ⁢                                                t                  mn                                ⁢                exp                ⁢                                                      {                                                                                            -                          j                                                ⁢                                                  xe2x80x83                                                ⁢                                                  k                          ⁡                                                      (                                                                                          m                                ⁢                                                                  xe2x80x83                                                                ⁢                                                                  δ                                  x                                                                ⁢                                X                                                            +                                                              n                                ⁢                                                                  xe2x80x83                                                                ⁢                                                                  δ                                  y                                                                ⁢                                Y                                                                                      )                                                                                              f                                        }                                    .                                                                                        (        12        )            
This equation means that the diffraction W(X,Y) on the image plane is a product of the sinc functions and the Fourier transform of tmn. Sinc functions appear in the diffraction from slits with definite apertures. Sinc functions do not appear in the diffraction from slits with infinitesimal apertures. In diffraction, the 0-th order diffraction is the strongest, the 1 st order diffraction is the next strongest and the 2nd order diffraction is the third strongest. The decrease of the higher order diffraction is well expressed by the sinc functions. A conventional diffraction grating included a single sinc function due to its one dimensional character. However, since DOEs are two-dimensional devices, W(X,Y) of DOEs include two sinc functions of x- and y-coordinates. The reason why the diffraction W(X,Y) is a Fourier transform of tmn is that the converging lens Fourier-transforms the diffracted waves. Namely, the function of the lens is the Fourier transform.
When the desired diffraction pattern on the image plane is continual one, tmn should directly be determined from the desired continual pattern from Eq. (12).
Discrete diffraction light distribution is more covenient for DOEs than analog diffraction distribution, since all the calculation is done by a computer. Furthermore, if the calculation is based upon the Fast Fourier Transform (FFT), the amplitude distribution on a DOE must has the same cell size as the diffraction distribution on the image plane. Since the unit of the DOE is briefly called a xe2x80x9ccellxe2x80x9d, a unit on the image plane will be named xe2x80x9cimage cellxe2x80x9d for discriminating the cell on the image plane from the cell of the DOE. The image cell is addressed by a horizontal number xe2x80x9cpxe2x80x9d and a vertical number xe2x80x9cqxe2x80x9d instead of X and Y. It is convenient to rewrite the inner variables to 2xcfx80 (m/M)p which is suitable to Fourier transform. The replacement requires 2xcfx80 (m/M)p=km xcex4xX/f in the x-direction. Namely, p=Mxcex4xX/f xcex. This determines the unit size xcex41 of an image cell in the x-direction as xcex41=fxcex/xcex4x M. X=pxcex41 is the relation between continual X and discrete p. Similarly, in the y-direction, q=Nxcex4yY/fxcex. The image cell unit length in the y-direction is xcex42=fxcex/xcex4yN. Y=qxcex42. Such quantization changes W(X,Y) into a discrete expression.                               W          pq                =                              δ            x                    ⁢                      δ            y                    ⁢          sin          ⁢                      xe2x80x83                    ⁢                      c            ⁡                          (                              p                M                            )                                ⁢          sin          ⁢                      xe2x80x83                    ⁢                      c            ⁡                          (                              q                N                            )                                ⁢                                    ∑                              m                =                                                      -                    M                                    /                  2                                                                              M                  /                  2                                -                1                                      ⁢                          xe2x80x83                        ⁢                                          ∑                                  n                  =                                                            -                      N                                        /                    2                                                                                        N                    /                    2                                    -                  1                                            ⁢                                                t                  mn                                ⁢                exp                ⁢                                                      {                                                                  -                        2                                            ⁢                                              xe2x80x83                                            ⁢                                              πj                        ⁡                                                  (                                                                                    mp                              M                                                        +                                                          nq                              N                                                                                )                                                                                      }                                    .                                                                                        (        13        )            
Tpq denotes discrete Fourier transform (DFT) of tmn.                               T          pq                =                              1            MN                    ⁢                                    ∑                              m                =                                                      -                    M                                    /                  2                                                                              M                  /                  2                                -                1                                      ⁢                          xe2x80x83                        ⁢                                          ∑                                  n                  =                                                            -                      N                                        /                    2                                                                                        N                    /                    2                                    -                  1                                            ⁢                                                t                  mn                                ⁢                exp                ⁢                                                      {                                                                  -                        2                                            ⁢                                              xe2x80x83                                            ⁢                                              πj                        ⁡                                                  (                                                                                    mp                              M                                                        +                                                          nq                              N                                                                                )                                                                                      }                                    .                                                                                        (        14        )            
A computer can easily calculate Tpq when tmn, M and N have been known. Using Tpq, the diffraction amplitude Wpq of the (p,q)-th image cell on the image plane is given by
Wpq=Mxcex4xNxcex4y sinc(p/M) sinc(q/N)Tpq.xe2x80x83xe2x80x83(15)
Diffraction intensity distribution Ipq on the image plane is the square of the amplitude Wpq.
Ipq(Mxcex4x)2(Nxcex4y)2 sinc2(p/M) sinc2(q/N)|Tpq|2.xe2x80x83xe2x80x83(16)
Tpq can be calculated from tmn by the fast Fourier transform. Tpq leads to the diffraction intensity Ipq at the (p,q)-th image cell.
A DOE is an element for splitting the incidence light spatially into a plurality of partial waves having definite power ratios. A DOE itself divides the incidence light in different directions without making images. For example, the DOE diffracts strong light with intensity 1 in some directions but diffracts no light with intensity 0 in other directions. In this case, the intensity of the diffraction is binary (two-values of 0 or 1). Of course, four-values or eight-values of diffraction intensity are also available. The design of a DOE is similar to any steps of diffraction intensity. The following explanation relates only to the binary intensity steps for simplicity.
The DOE lacks the function of convergence. The diffracted partial waves are parallel for indefinite long distances. In practical use, parallel diffracted waves are converged by a converging lens on an image plane. The image pattern made by the lens has bright regions having strong diffraction intensity xe2x80x9c1xe2x80x9d and dark regions having no diffraction intensity xe2x80x9c0xe2x80x9d. The bright regions having diffracted light are now named xe2x80x9csignalxe2x80x9d regions. Image cells in the signal regions are called xe2x80x9csignal image cellxe2x80x9d. The other regions in which no light should be diffracted are named xe2x80x9cblankxe2x80x9d regions. Image cells in the blank regions are called xe2x80x9cblankxe2x80x9d cells. It is desirable that the diffraction power supplied into the signal cells should be equal. The fluctuation of the diffraction intensity in the signal cells is called xe2x80x9cintensity non-uniformityxe2x80x9d. Smaller intensity non-uniformity is more desirable. The intensity that appears in blank cells is called xe2x80x9cnoisexe2x80x9d. Less noise is preferable.
The diffraction efficiency xcex7 can be another parameter constituting a merit function. The diffraction efficiency is defined as a ratio of the energy on the signal cells to the whole incidence energy. The diffraction efficiency is an important property. Design of a DOE aims at a larger diffraction efficiency, a smaller intensity non-uniformity and less noise. Then, objectives are predetermined as ideal values for the characteristic properties, i.e., diffraction efficiency, non-uniformity and noise. The DOE design tries to optimize the phase distribution of the DOE for bringing the characteristic properties close up to the objectives. What brings the characteristic properties close to the objectives is a merit function. The merit function shall include the differences between the important properties and their objectives in a relation in which the merit function should diminish, when the characteristic properties approach their objectives. A simple definition of a merit function is a sum of squares of the differences between the properties and the objectives. This definition enables the merit function to decrease in accordance with approaches of the properties to the objectives. However, other properties excluded from the merit function are intangible. The merit function is neutral and indifferent to the other properties which are not contained. The non-selected properties are out of control for the merit function.
Here, the diffraction efficiency, the non-uniformity and the noise intensity are chosen as characteristic properties which shall be contained in the merit function.                     E        =                                            W              1                        ⁢                          1                              N                s                                      ⁢                                          (                                                      η                    obj                                    -                  η                                )                            2                                +                                    W              2                        ⁢                                          ∑                                                      (                                          p                      ,                      q                                        )                                    ∈                  s                                            ⁢                              xe2x80x83                            ⁢                                                (                                                            I                      av                                        -                                          I                      pq                                                        )                                2                                              +                                    W              3                        ⁢                          N              s                        ⁢                                                            Max                                                            (                                              p                        ,                        q                                            )                                        ∉                    s                                                  ⁡                                  (                                      I                    pq                    2                                    )                                            .                                                          (        17        )            
Here, W1, W2, and W3 are constant weights, Ns is the number of the signal cells. S is the signal regions. xcex7 is the diffraction efficiency for the given parameters. xcex7obj is an objective of xcex7. Ipq is the diffraction intensity at the (p,q) image cell within the signal regions. Iav is an average intensity of the diffraction signals in the signal regions. The first term is a square of the deviation of the diffraction efficiency from the objective. The second term is a sum of the squares of the differences between the individual intensity Ipq and the average intensity Iav in the signal regions. The last term denotes noise intensity. The last term is the maximum of the squares of the noise intensity Ipq of the blank cells (p,q) outside of the signal regions S. Ideally, no light should be diffracted to the blank cells, but some is diffracted also to the blank cells as noise. The noise is estimated here by the maximum intensity instead of the average or the sum.
This merit function would diminish to 0, if there were no noise, the diffraction intensity were uniform for all the signal cells and the diffraction efficiency were equal to the objective. In general, when the noise decreases, the signal intensity is more uniform or the diffraction efficiency approaches the objective, the merit function decreases. The merit function will be changed to a minimum by varying the variables of the DOE. The set of variables minimizing the merit function yields the least noise, uniform signal intensities and objective diffraction efficiency. Minimizing the merit function brings about a pertinent set of variables.
The DOE design requires an optimizing calculation of minimizing the merit function which includes phases of tens of thousands to millions of cells as variables. An immense number of cells sometimes demands a long calculation time. FIG. 24 denotes the steps of design of a DOE in binary phase case. At first, initial values for phases of DOE cells are given. The phase of each cell is 0 or xcfx80. xcfx86mn is the phase for the (m,n)-th cell. xcfx86mn is either 0 or xcfx80. In the example, initial phases are given to all the cells at random. Besides the random initial phases, initial phases can be determined by some rule.
The assignment of the initial phases leads to the DOE-diffracted intensity distribution on the image plane through Eq. (6) to Eq. (16). The merit function is calculated for the diffraction pattern. A number r which starts from 1 is affixed to the merit function E into Er The first merit function is E1. Then, phase of a cell (u,v) is reversed. If the current phase of the (u,v)-th cell is xcfx80, the phase should be changed into 0. Otherwise, if the current phase is 0, it is changed to xcfx80. The other cells (MN-1) maintain their previous phases. The new set of phases determines again the DOE-diffracted intensity distribution {Ipq} on the image plane. {Ipq} gives a new value Exe2x80x2 for the merit function. Exe2x80x2 is compared with E. If Exe2x80x2 is smaller than E, the phase reversion of the (u,v) cell is accepted. If Exe2x80x2 is not smaller than E, the phase reversion is rejected. The phase of the (u,v)-th cell is restored. The merit function keeps the previous value E.
The order of the cells which reverse the phase can be arbitrarily determined. This example starts from u=0 and v=0. The phase-changing cell transfers to the right from (0,0) to (1,0), (2,0), . . . , and (M-1,0) one by one. When u arrives at M (u=M), the alteration of phases finishes for the line v=0 and moves to the second line (v=1). When the phase-alteration finishes at the (Mxe2x88x921, v) cell for line v, it transfers to the first cell (0, v+1) of the next line v+1. Every alteration of a phase is followed by the calculation of {Ipq} and the merit function E. The phase-alteration lowering the merit function shall be accepted but the phase-alteration raising the merit function shall be abandoned. When the phase alteration arrives at the lowest, right most cell (u=Mxe2x88x921, v=Nxe2x88x921), the merit function E has been reduced by some amount from the initial merit function E1. The decrement (E1xe2x88x92E) from the initial E1 to the final E is calculated. When the decrement is larger than a critical value xcex5(E1xe2x88x92E greater than xcex5), there is still a probability of reducing the merit function by the phase-alteration. Then, the current E replaces E1(Exe2x86x92E1). The alteration of the phases shall be again repeated from the uppermost, leftest cell (0,0), similarly to the former procedure. What shall be repeated is the steps of altering the phase of the (u,v)-th cell, calculating the diffraction intensity distribution Ipq, calculating the merit function comparing the new merit function with the previous one, replacing the phase of the (u,v)-th cell when the new merit function is smaller than the previous one, or rejecting the phase alteration when the new merit function is not smaller than the previous one. The serial phase-alteration step should be repeated for reducing the merit function E. Some repetitions of the serial phase alteration step bring the merit function to a minimum, (0xe2x89xa6Erxe2x88x92E less than xcex5). Then, the value Er is the minimum value of the merit function which can be attained from the given initial set of phases which have been randomly settled. When Erxe2x88x92E less than xcex5, the calculation of r=1 shall be ended. The calculation brings about a set of phases xcfx86mn based upon the r=1 intial values and a minimized merit function E.
However, the minimum Er depends upon the initial phases. The minimum Er is not necessarily the absolute minimum of the merit function.
FIG. 25 shows the relation between a parameter and a merit function. The abscissa is a parameter which represents many parameters in brief. The ordinate is the merit function. The parameter varies the merit function. When I1 is chosen as an initial point, the phase-alteration step will bring the merit function to a bottom A at xa. E stops at point A. When another initial point I2 is selected, the merit function will fall to another minimum B at xb by the phase-alteration step. When a further initial point I3 is chosen, the merit function can attain a minimum C at xc. The minima depend upon the initial parameters. It is necessary to start from various initial values, calculate minima from the initial values and seek the least minimum among the minima. In FIG. 25, the merit function should take point C as the absolute minimum, abandoning points A and B.
The process should be returned to the initial phase settlement (xcfx86mn=0 or xcfx80 at random). Another set of initial phases is again given at random. A similar calculation shall be repeated from the second set of initial values. The repetition of altering the phase, calculating a merit function and replacing the phase leads to a minimum merit function E2 which can be attained from the second set of initial values. Aplurality of sets of initial values are given. The current (r=2) minimum merit function Eopt shall be compared with the previous (r=1) minimum merit function. When the current (r=2) minimum merit function is smaller than the previous (r=1) minimum merit function, the current (r=2) set of the merit function Eopt, the phases xcfx86mn shall be accepted. Otherwise, if the current merit function is bigger than the previous merit function, the current Eopt, the phases xcfx86mn shall be rejected. The same processes shall be repeated. The times of setting initial values are predetermined to R-1. When the renewal of initial values attains to R times, the calculation of minimizing the merit function shall be ended. The least merit function is the minimum of {E1, E2, . . . , Er}.
FIG. 26 shows the steps of making a DOE. xe2x80x9cPattern designxe2x80x9d seeks the optimum phase pattern on the DOE for the designated purpose. The processes described so far relate to the pattern design. This invention aims at an improvement of the step of the pattern design. The pattern design is followed by microprocessing, coating and inspection. The following three processes are beyond the scope of the present invention. Since the pattern design has determined the thicknesses (or heights) of all the cells, a flat substrate is microprocessed for making steps of cells.
FIG. 27 denotes the microprocessing. A DOE substrate is a flat board which is transparent for the laser light. For example, a ZnSe substrate is employed for a CO2 laser. A photoresist is coated on the substrate and prebaked. The photoresist can be either a positive type or a negative type. FIG. 27 shows the case of the positive type resist. A latent pattern is depicted on the photoresist by placing a photomask having desired mask pattern, shooting the photomask with ultraviolet lamp (UV) and exposing the photoresist. The black parts of the mask shield the UV light. The regions beneath the black parts are free from the UV light. The transparent parts allow the UV light to irradiate the resist. The UV light composes photoacid reaction in the resist and the photoacid reaction breaks polymer couplings. Development eliminates the resist of the UV-irradiated regions but leaves the other parts shielded by the mask untouched. The substrate partially-covered by the resist is treated by anisotropic etching. The etching step perforates cavities at the bare parts on the substrate. The resist is removed. The substrate possesses a binary stepped surface. The height of the step is xcex/2(nxe2x88x921) which corresponds to a phase difference of xcfx80. The bottom of the left column of FIG. 27 shows the binary step DOE. If a similar process is repeated for making new steps of xcfx80/4 and 3xcfx80/4, a four-stepped DOE can be produced, as denoted by the right column of FIG. 27. N-repetitions of the photoetching will make a 2N-stepped DOE. If laser light shot the naked substrate, the light would be reflected by the front surface and the rear surface, which would lower the diffraction efficiency. Then, both surfaces of the substrate are coated with antireflection films of dielectric multilayers. The produced DOE shall be inspected whether it exhibits the desired performance by irradiating the DOE board by laser light, measuring the diffraction power on the image plane, as shown in FIG. 22, and comparing the diffraction with the desired image pattern.
If a lens assembly were produced without production errors, the lens assembly would reveal the best performance that is determined by the solution. The best performance is called a designed performance. However, since production errors accompany an actual lens assembly, the performance of actual products is inferior to the designed performance. An estimation of lens assembly should take production cost and production feasibility into consideration as well as performance. The prior estimation method, however, has a drawback, since it has traditionally ignored the production feasibility. If the prior estimation method judged a solution to be optimum, it is not necessarily feasible to produce the lens assembly having the parameters determined by the solution. Actual products, in general, do not exhibit the designed performance.
Here, the word xe2x80x9cperformancexe2x80x9d should be defined. An optical part or a lens has many individual properties. An assembly of individual properties is the performance. Individual properties are measurable. But performance cannot be measured directly, since performance is defined as a set of properties. Inherently performance is not measurable. But in order to estimate a product by the performance, the performance should be converted from an immeasurable concept to a measurable value. Since performance is an assembly of properties, the performance can be changed to a measurable value by expressing the performance as a sum of individual properties with weights. The summation of properties endows the performance with a new character as a measurable value. However, unless the weights are determined, the summation is not carried out. The judgement of the importance of properties decides the weights case by case. Then, assuming the weights have been applied to performance, the performance is a measurable, integral variable for estimating the product.
Individual properties depend upon the parameters and the variables of a lens assembly. There are many variable parameters in a lens assembly. Lens thicknesses, curvatures of front surfaces and rear surfaces, aspherical coefficients of surfaces, distances between lenses and so forth can be variables. xe2x80x9cDesignxe2x80x9d is a process of determining optimum values of the variables. The values of variables which satisfy the equations are called a xe2x80x9csolutionxe2x80x9d. A parameter has a definite value for satisfying the equations. A set of the definite values makes a solution.
An optical part has an object. Equations are set up for seeking the parameters accomplishing the object. However, the number of the equations is often fewer than the number of the variable parameters. The equations cannot give a unique solution. Approximating calculation bears a plurality of solutions which satisfy the set of equations. Namely, lack of the restrictions makes many solutions. Then, the design does not end by finding a solution at all. The most suitable one should be selected from a plurality of solutions whose number may be indefinite. The equations cannot determine the most suitable solution. The most suitable solution should be determined rather from standpoints different from the optical equations.
The merit function is just the means for selecting the most suitable solution from many candidate solutions. The merit function is a sum of squares of ray aberration or optical path difference at individual points. A bigger merit function signifies worse performance. A smaller merit function means better performance. At present, the estimation of aberration is the main purpose of the merit function. Namely, if a plurality of solutions are found, the solution which gives the smallest aberration should be chosen by the merit function. Lowest aberration solution is the unique criterion for the solutions at present. The aberration is a decline of performance of a produced one from the ideal one. It may be reasonable to determine the validity of solutions by aberration. The current method chooses the lowest aberration solution as the most suitable solution from many candidate solutions. This is an intuitive and primitive estimation.
However, any prior estimation has never been based upon the feasibility of production. Conventional merit functions have ignored the feasibility of production. Such a selection has a drawback. If a lens assembly having the parameters which are all equal to the values given by the optimum solution were to be built, the lens assembly would exhibit excellent properties. However, production errors surely accompany actual manufacturing. The production errors degrade the performance of products below the ideal performance. Sometimes a production error for some parameter is fatal and a small error of the parameter degrades the performance to a great extent. Much attention should be paid to such a dangerous parameter for suppressing the production error. In spite of the keen attention, some production errors arise in any case. Then the products should be examined. Some of the products should be abandoned as rejected articles, if they have large errors beyond the tolerances for some parameters. Designs with small tolerances for dangerous parameters are suffering from low yield and high production cost.
FIG. 4 is an imaginary graph showing the dependence of performance upon the values of a parameter. There are many parameters having an influence upon the performance. But one parameter is now adopted in the graph as an abscissa. The performance, which is inherently an abstract character, is assumed to be measurable and is denoted in a reverse direction along an ordinate. A search for an optimum solution by using a merit function begins with an adoption of initial values of parameters. For example, an initial value is taken at point-xcex2 in FIG. 4. Then the parameter is changed to the right on the curve for raising the performance to point-xcex3. Since point-xcex3 gives the merit function a minimum value, point-xcex3 gives the value of the parameter in a solution. But the solution depends upon how to take an initial value. If someone starts from another initial point-xcex4, he will vary the parameter little by little to the right till the bottom point-xcex5. This is the value of the parameter in another solution. Another initial point-xcex8 leads to a solution at point-xcex7. The example of FIG. 4 shows three solutions depending upon the initial values. Owing to an excess number of variables and shortage of confining conditions, the judgement by the minimum merit function bears a lot of solutions. The solutions depend upon the initial values. Then, what is important is the selection of the initial values of variable parameters. We don""t know yet how to find the best initial values.
Besides the selection of the initial values, there is another problem. The problem is that the performance is not the unique factor that decides what solution is the most suitable one. This is more important but is more difficult to understand problem. Point-xcex5 is superior to point-xcex7 in performance in the example of FIG. 4. Prior estimation would choose point-xcex5 as an optimum solution.
However, point-xcex5 lies at a very narrow valley. If the design were to be done on the solution based upon point-xcex5, the performance would fall rapidly due to a small deviation (error) from point-xcex5. Point-xcex5 is a dangerous minimum. On the contrary, point-xcex7 is inferior to point-xcex5 in the ideal performance. Since point-xcex7 lies in a wide valley, the fall of the performance around point-xcex7 is far smaller than that around point-xcex5 to the same errors. Point-xcex7 is a safer minimum than point-xcex5. Further an initial point-xcex2 leads to a valley of point-xcex3. At point-xcex3, the relation between the fall of the performance and the error is looser than point-xcex7. In practice, production errors surely accompany production of a lens assembly. If a lens assembly is designed at point-xcex5, the rapid fall of the curve prevents manufacturers to make a product just having the designed performance. On the contrary, if point-xcex7 is selected as a designed value, the production is easier than point-xcex5, since the tolerance has a larger margin at point-xcex7 than at point-xcex5. Namely, the small tolerance makes the production based upon point-xcex5 difficult. The large tolerance endows feasibility to the production based upon point-xcex7 or point-xcex3. Point-xcex7 and point-xcex3 are superior to point-xcex5 from the standpoint of feasibility of production.
The finally-attained optimum values (solution) depend upon the initial, starting values. However, the initial values are not a unique factor for determining the final optimum values. The final solution depends also on the order how to change the parameters from the initial values. Different orders of changing parameters lead to different minimum values (different solutions). There is a freedom of selecting initial values. The order of changing the parameters has also a freedom. The guideline of minimizing the merit function cannot necessarily lead us to a solution having wide tolerances. If the solution obtained by minimizing the merit function has poor tolerances, it is hard to make the product in accordance with the solution.
FIG. 5 exhibits the relation between a production error and performance schematically. An optimum value x0 of a parameter accomplishes the best performance (designed performance). The abscissa is a production error of the parameter which is the difference between the actual value x and the optimum value x0. The ordinate means the performance. If the error is zero, the product will exhibit the best performance at point-xcex. The performance falls in proportion to an increase of the production error in both directions. A horizontal line xcexaxcexc means the minimum allowable performance (standard performance). The errors at xcexc and at xcexa determine the tolerance xcex94 of the production error. The error at point xcexc is the plus tolerance (+xcex94). The error at point xcexa is the minus tolerance (xe2x88x92xcex94). The tolerances (xc2x1xcex94) are equal to the errors at point xcexa and xcexc which give the minimum allowable performance. The lines xcexaxcex and xcexxcexc are allowable scope of the performance and the production errors. When a product shows the performance of point xcexd with an error e1 (e1 greater than xcex94), the product should be rejected. The performance depends upon many parameters and variables which have inherent tolerances. Some parameters have wide tolerances. Other parameters have narrow tolerances. Attention should be paid to the parameters with narrow tolerances. A small deviation e=(xxe2x88x92x0) of the parameters from the optimum value degrades rapidly the performance. The narrow tolerance requires a careful operation for the parameter. In spite of the carefull operation, products are suffering from an error which is larger than the tolerance. The products having a quite small error which is yet larger than the tolerance should be rejected. Namely, narrow-tolerance parameters lower the yield by increasing the difficulty of production.
An imaginary new concept of dS/de which is a ratio of the fall of performance to an increment of an error may help the understanding of the relation between the difficulty of production and the tolerance of errors. A large dS/de is undesirable in actual manufacturing, because only a small error degrades the performance of products fatally. The estimation based only upon aberration notifies us about nothing of the influence of production errors on products. Then, after determining the optimum solution of parameters, allowable widths of errors, that is, tolerances, are individually allotted to the optimum parameters for clarifying the guidelines of production. The difficulty of production is totally different between a 10 mm thick lens of a 100 xcexcm tolerance and another 10 mm thick lens of a 3 xcexcm tolerance. It is far harder to produce the latter one of the quite small tolerance than the first one.
When an optimum solution has the parameters having a quite small tolerance, the production in compliance with the optimum solution is difficult. Prior estimation of designs of optical parts has entirely lacked the viewpoint of estimating the designs by the degrees of the difficulty of actual production. Conventional estimation of the designs has not adopted wide-tolerance solutions which facilitate actual production but has adopted low-aberration solutions which realize high performance. The estimation of the present invention is entirely different from the conventional ones.
Unlike prior estimation, this invention gives a new estimation which can estimate the difficulty of the production based on solutions and can produce an optimum solution having parameters with wide tolerances which ensure easy production.
Conventional DOE design has sought optimum phase pattern of a DOE by considering a non-error state, establishing a merit function for the non-error state and minimizing the merit function by changing phases. The prior DOE design has taken no account of production errors. However, errors always accompany production. Accidental production errors lower the performance of the DOE. The error-induced degeneration has not included in the conventional DOE design. Admitting the possibility of degeneration by production errors, prior design has believed that the degradation can be prevented by lowering errors and has tried to suppress the production error as low as possible.
FIG. 28 is a figure showing prior art DOE design. First of all, various kinds of restrictions are settled. For example, the pattern size is determined in accordance with the purpose of the DOE. The pattern size is, e.g., 64 cellsxc3x9764 cells, 128 cellsxc3x97128 cells or so. The initial cell number should be determined by considering the light source power, the size of the image plane, the complexity of the object pattern. The number of steps of phases is also a parameter determined arbitrarily. The simplest phase step is a binary phase step consisting of 0 and xcfx80. Here the phase is equivalent to the thickness or the height to cells of the DOE. In the case of transparent type DOEs, the phase difference of a step is 2xcfx80(nxe2x88x921) xcex94t/xcex, where xcex94t is the height of the step, n is the refractive index and xcex is the wavelength of the light. The binary steps of the phases of 0 or xcfx80 should have the difference xcex94t=/2(nxe2x88x921) in height. Since diffraction arises from periodically aligning binary steps, the binary DOE is available. More sophisticated diffraction patterns require higher degrees of steps, i.e., four degree steps (quadruple), eight degree steps or so. A quadruple DOE allocates phases 0, xcfx80/2, xcfx80 or 3xcfx80/2 to cells. The top surfaces of cells take four degrees of heights. The least difference between the cell heights is xcex94t=xcfx80/4(nxe2x88x921). A DOE has a simple structure of a matrix of cells which align lengthwise and crosswise. The simple structure forbids the DOE from having many restrictions. Restrictions are the cell size, the cell number and the step number. The refractive index is predetermined by the material of the DOE. Thus, the refractive index is not an arbitrarily-determined restriction.
There is also a freedom how to build up the merit function. A usual manner has been used to construct a merit function by summing up the squares of the differences between the calculated parameters and their objectives. The definition can equate a decrement of the merit function with the approach of the parameters to the objectives. For example, diffraction efficiency, intensity fluctuation, noise or so can be selected as the properties(parameters) included in the merit function. Actually, the diffraction efficiency can be included in the merit function as a square of the difference between the calculated efficiency and the predetermined objective. The intensity fluctuation means the dispersion of the light power attaining to the plural cells which should receive the diffracted light. The merit function can contain the intensity fluctuation as a sum of squares of differences between the intensity entering individual signal cells and the average intensity of them. Noise is the light entering the blank cells to which the light should not be diffracted. No noise is an ideal limit. But noise somewhat accompanies an actual DOE. Thus, the squares of noise intensity of the blank cells are included as the form of either a sum or a maximum of the squares in the merit function for decreasing noise. Since noise intensity is always positive, the noise power itself can be included in the merit function instead of the square. Other properties can also be included in the merit function as a square of differences between the values of properties(parameters) and their objectives. Namely, the object diffraction pattern determines the structure of the preferable merit function. Pertinent choice of the merit function can realize the target object diffraction pattern.
An optimizing calculation follows the determination of the restrictions and the merit function. For example, it is assumed that a single unit should have 64 cellsxc3x9764 cells having a binary phase (0 or xcfx80). First of all, initial phases should be assigned to all the cells. There are a lot of sets of initial values assigned to the cells. The minimum merit function attainable from the initial values depends upon the choice of the initial values, as clarified by referring to FIG. 25. Initial phases may either be given at random without considering the object pattern or suitably by considering the object pattern. For example, even the simplest case of the unit of 64 cells by 64 cells with binary phases has 24096 candidates for phases of cells. One of them is chosen as an initial set of phases. The initial phases enable a designer to calculate the diffraction pattern on the image plane. The image pattern is also quantized into image cells on the plane. The image pattern is an assembly of image cells and the diffracted power on the image cell. The merit function can be calculated from the image pattern. Then, a phase of a cell is altered. The diffraction pattern is calculated. The merit function is again calculated. If the new merit function is lower than the previous one, the phase alteration should be confirmed. Otherwise if the new merit function is equal or bigger than the previous one, the phase should be restored. Repetitions of the phase alterations decrease the merit function to a minimum value. This is only a bottom value that can be attained from the initial phases. The minima depend upon the initial set of phases. Thus, the initial phases are fully changed. Similar steps are repeated from the new set of the initial phases for seeking another minimum of the merit function. Several number of sets of the initial phases determine the same number of minima of the merit function. Among the minimum estimation values, the smallest value should be elected as a suitable merit function giving a solution having appropriate properties. These processes are the operation of minimizing the merit function in FIG. 28.
However, the solution is not a decisive one until it is confirmed that the solution satisfies the desired performance. The solution shall be further investigated from two standpoints. One point is whether the solution satisfies the desired conditions (properties, performance). The other point is tolerance analysis. This problem may be rather difficult to understand. Even if an imaginary DOE having just the parameters of the solution satisfies the desired performance, an actual DOE manufactured after the solution sometimes does not satisfy the desired performance owing to production errors. Errors accompany production. The degradation due to production errors is important. It is necessary to investigate the degradation caused by the production errors. When production errors are small, the performance of the DOE still satisfies the required performance. The production errors are allowable errors. Further increase of production errors equalizes the performance to the minimum performance. The production errors are the maximum of the allowable errors. The maximum of the allowable production errors is called tolerance. When all the parameters exist within the scope of the designed valuexc2x1tolerance, the DOE reveals sufficient performance. If some of the parameters exist out of the scope of the designed valuexc2x1tolerance, the DOE cannot satisfy the required performance.
It is facile to produce good DOEs when tolerances are enough wide. But when tolerances are small, it is difficult to produce good DOEs having the parameters within designed valuesxc2x1tolerances. When some tolerances of the solution are too small to contain the parameters within the scope of the designed valuesxc2x1tolerances, the solution should be rejected as improper one. Although the solution has been calculated by minimizing the merit function after a long calculation, the solution should be abandoned. The current step should return to the beginning step of setting restrictions.
More complicated restrictions should be imposed for improving the calculation based upon the merit function. For example, a pattern of a DOE shall be converted from 64 cellsxc3x9764 cells to 128 cellsxc3x97128 cells. Otherwise, the steps of phases shall be increased from binary steps to quadruple steps. A similar cycle of steps shall be started under the renewed and complicated restrictions. Namely, a determination of initial phases is followed by the steps of calculating the diffraction pattern, deriving the merit function, altering a phase of a cell, calculating diffraction pattern, deriving the merit function, . . . and obtaining a minimum merit function. Then, another determination of initial phases begins the same cycle of calculations. The least of the minimums of the merit functions is then determined under the restrictions either the unit of 128 cellsxc3x97128 cells or the quadruple phases. The new solution should be tested by the two standpoints as mentioned once. First, it should be confirmed whether the parameters of the solution satisfy the desired performance. Second, the maximum allowable errors, tolerances, should be sought by surveying the degeneration induced by production errors. If the tolerances are wide enough, the solution should finally accepted. But if the tolerance is not so wide yet, the solution should again be rejected and the current process should return to the first step of FIG. 28 three times. Prior art design used to seek the optimum solution of the optimum parameters by repeating the steps of FIG. 28 until all the parameters obtain sufficiently large tolerances.
The prior art design believes that the replacement of coarse restrictions by fine restrictions should improve the performance of the DOE. At first, the prior method minimizes the merit function under the restriction of a unit size of 64 cellsxc3x9764 cells. If the solution is denied by the result estimation step, the prior design further elaborates the restriction by increasing the number of a unit to 128 cellsxc3x97128 cells. The increment of the cell number per unit raises the amount of calculation but will enhance the performance of the DOE. Since the merit function is a sum of squares of errors, the elaboration of restrictions will improve the performance of the DOE.
The situation is, however, entirely different in tolerances. The merit function does not include tolerances. It is uncertain whether the solution obtained by minimizing the merit function free from tolerances would bring about wide tolerances or not. The merit function is fully irrelevant to tolerances. Prior design does the tolerance analysis and returns the processing to the initial step of determining the restrictions, if the tolerance turns out to be too narrow. Elaborating the restrictions does not necessarily lead to increases of tolerances. For example, the elaboration from 64 cells to 128 cells may increase tolerances in some cases but may decrease tolerances in other cases. The merit function has no motive of raising tolerances. Minimizing the merit function does not ensure an widening of tolerances. The design based upon the prior merit function is useless to enhance tolerances.
Unfortunately, the upgrade of the restrictions, e.g., an increment of cell number or an increase of phase steps, sometimes tends to decrease tolerances. When a unit is increased from 64 cells to 128 cells, the cell size is reduced to a half, which has a tendency of decreasing the tolerance. The merit function has no function of enhancing tolerances. The upgrade of the restrictions has rather a tendency of lowering tolerances. The repetitions of the steps of FIG. 28 sometimes cannot enlarge tolerances up to desirable values. In this case, the repetitions of FIG. 28 cannot end and the computer must repeat insignificant operations. The inconvenience originates from the fact that the conventional merit function is interested only in the improvement of performance but is unresponsive to tolerances. The problem is widening tolerances.
One purpose of the present invention is to provide a method of design of a DOE which endows parameters with large tolerances. Namely, the purpose of the invention is facilitating the manufacture of DOEs by allowing large tolerances to the parameters.
The method of designing optical parts includes the steps of making a merit function E0 for the set of parameters without errors and extra merit functions E1, E2, . . . for the set of parameters with positively-allotted errors xc2x1xcex4, multiplying weights w0, w1, w2, . . . by the merit functions E0, E1, E2, . . . , making an integrated (unifed) merit function E=w0E0+w1E1+w2E2+ . . . =xcexa3wkEk and seeking a set of parameters which minimize the integrated (unified) merit function.
The conspicuous feature is the initial positive allotment of errors xc2x1xcex4 to the parameters for building the integrated (unified) merit function. The error-allotted parameters should be selected as the parameters which are difficult to adjust to the designed values. The error which is initially allotted to an object parameter is called an allotted error xc2x1xcex4 for discriminating the production error e or the tolerance xcex94. The allotted error is the main concept of the present invention. The allotted error xc2x1xcex4 should be selected to be larger than the ordinary production error e. Three kinds of errors should be discerned. The production error accompanies an actual manufacture of a product. The production error is unintentional, is different for individual products, takes many values for a single parameter and is a probability variable. The tolerance is a definite value determined uniquely to each parameter as a maximum of allowable production errors, and has a single value for a single parameter. The allotted error is an error which is initially allocated to a single selected parameter or a few selected parameters. The allocated error is a single value to the selected parameter. The rest of the parameters have no allotted error.
This invention assumes a plurality of error-allotted states S1, S2, S3, . . . which include one parameter (or two or three parameters) allotted with an allotted error. The error which is positively given to the parameter is called an allotted error for discriminating the production error. The state including the parameters having the allotted error is called an error-allotted state. The original state S0 without the allotted-error is called a non-error state. The sum E=xcexa3wkEk of the weighted merit functions is called a unified or integrated merit function.
FIG. 12 shows the flow of the method of the present invention. State S0 is the non-error state. E0 is the merit function of state S0. State S1 is a state allocating a plus error +xcex4 to a selected parameter Pi. E1 is the merit function of the error-allotted state S1. State S2 is a state allocating a minus errorxe2x88x92xcex4 to the same selected parameter Pi. E2 is the merit function of the error-allotted state S2. Pi is the selected parameter. The selected parameter Pi takes different values Pi, Pi+xcex4, and Pixe2x88x92xcex4 for three states S0, S1 and S2. The other parameters Pj(jxe2x89xa0i) have a common value for S0, S1 and S2. Instead of a single parameter, two or three parameters can be chosen as selected parameters for being allocated with errors. In the plural error-allocated parameters, extra states S3, S4 . . . and merit functions E3, E4, . . . should be taken into account for estimating solutions besides S1 and S2. The unified merit function E=xcexa3wkEk can be obtained by determining the values of all variables, calculating wavefront aberration or ray aberration, calculating the merit functions E0, E1, E2 . . . and summing w0E0, w1E1, w2E2, . . . up to the unified merit function. The merit function can be calculated by determining a set of variables. The next step is minimizing the merit function E. A set of values of variables corresponds to a value of the merit function. The set of variables which give the smallest value to the merit function is sought. The set of variables which realizes the minimum merit function should be taken as an optimum solution. This is the minimizing calculation (optimization). The minimum of the merit function finally determines the set of optimum variables. In practice, the minimizing process can be carried out by setting initial values of all variables, calculating aberration (wavefront aberration or ray aberration) at all relevant points and calculating the merit function again. If the merit function is reduced by the change of the variable, the change of the variable is adopted (accepted). If the change of the variable enhances the merit function, the change of the variable is abandoned (rejected). The reduction of the merit function determines the direction of the change of variables. The smallest merit function is sought by changing all the available variables in succession.
When optimum values of variables are determined by minimizing the merit function, the result is further estimated. The estimated result includes the performance analysis and the tolerance analysis (tolerancing). Since this invention initially gives allotted-errors to some parameter, the error-allotted parameter will allow wide tolerance. The large tolerance ensures easy manufacture of the product. The large tolerance enhances the productivity and lowers the production cost.
This invention assumes error-allotted states which include errors for some parameters, calculates merit functions of the error-allotted states, adds the merit functions of the error-allotted states to a merit function for a non-error state for making an integrated merit function and minimizes the integrated merit function for determining optimum parameters. It is preferable to produce the error-allotted states to the parameters which require large tolerances. FIG. 29 demonstrates the DOE design of the present invention. State S0 is an ideal non-error state free from errors. E0 is a merit function for S0. S1 is an error-allotted state which supplies a selected parameter Pi with +xcex4i a priori. E1 is a merit function of S1. S2 is an error-allotted state which allocates the parameter Pi with xe2x88x92xcex4i. E2 is a merit function of S2. In addition, other states S3, S4 . . . can be produced as error-allotted states which yield another parameter Pj with xc2x1xcex4j. Instead of allotting xc2x1xcex4 to the same parameter, errors +xcex4j, xe2x88x92xcex4k can be allotted to different parameters of Pj and Pk. The design of the present invention creates merit functions E1, E2, E3, E4, . . . of error-allotted states in addition to a merit function E0 of a non-error state S0, produces an integrated merit function E by summing up the merit functions E1, E2 . . . together with E0, minimizes the integrated merit function and determines a set of optimum parameters. The inclusion of error-allotted merit functions characterizes this invention.
In FIG. 29, the uppermost rectangles show a non-error state S0, an error-allotted state S1 giving Pi with +xcex4i, another error-allotted state S2 supplying Pi with xe2x88x92xcex4i and so on. These states have merit functions E0, E1, E2, . . . An integrated merit function E can be made from the individual merit functions by multiplying weights wk by Ek and summing up wkEk.                     E        =                              ∑            k                    ⁢                      xe2x80x83                    ⁢                                    W              k                        ⁢                                          E                k                            .                                                          (        18        )            
Minimization of the integrated merit function gives a solution of optimum parameters. Except for the use of the integrated merit function, this invention takes similar steps to prior methods. The DOE design includes the repeated steps of assuming initial values of parameters, calculating the merit function, altering phases, calculating the merit function, comparing the current merit function with the most recent one, and accepting the phase alteration when the merit function falls in value but rejecting the phase alteration when the merit function value rises, for minimizing the integrated merit function. The minimum merit function gives a solution of optimum parameters. Then, the solution is examined to determine whether the solution satisfies the requirements of performance and tolerances, as shown in FIG. 28. Thus, this invention implements a novel step of minimizing the merit function in FIG. 28. Instead of E0 of the non-error merit function, this invention takes account of error-allotted merit functions E1, E2, . . . . The gist of the present invention resides in the replacement of E0 by xcexa3wkEk.
This invention seeks an optimum set of lens parameters by making a non-error allotted ordinary merit function E0, making some error-allotted states allocating errors to some parameters, producing an integrated merit function E=E0+E1+E2+E3+ . . . by summing up the merit functions, calculating the integrated merit function by changing parameters, minimizing the integrated merit function and determining the optimum parameters which minimize the integrated merit function. The tolerances for the parameters for which xc2x1xcex4 errors have been allotted are enhanced. The large tolerance alleviates the difficulty of production of the lens system.
This invention seeks the most suitable phase distribution of a DOE by considering error-allotted states, making merit functions for the error-allotted states, producing an integrated merit function including the error-allotted merit functions, and minimizing the integrated merit function. Taking account of the error-allotted states enables the optimization calculation to make large tolerances for the parameters which have been considered into the integrated merit function. Large tolerances increase the feasibility of the production of DOEs.